Gaussian integral means of entire functions: logarithmic convexity and concavity

For $0<p<\mathrm{\infty }$ and $\alpha \in (-\mathrm{\infty },\mathrm{\infty })$ we determine when the $L^p$ integral mean on $\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }\le r\}$ of an entire function with respect to the Gaussian area measure $e^{-\alpha \mathrm{\mid }z{\mathrm{\mid }}^2}dA(z)$ is logarithmic convex or logarithmic concave.